A note on the spin-embedding of the dual polar space \({DQ}^-(2n+1,\mathbb{K})\)

Abstract

In \([6]\), Cooperstein and Shult showed that the dual polar space \({DQ}^-(2n+1,\mathbb{K})\), \(\mathbb{K} = \mathbb{F}_q\), admits a full projective embedding into the projective space \({PG}(2^n – 1,\mathbb{K}’)\), \(\mathbb{K}’ = \mathbb{F}_{q^2}\). They also showed that this embedding is absolutely universal. The proof in \([6]\) makes use of counting arguments and group representation theory. Because of the use of counting arguments, the proof cannot be extended automatically to the infinite case. In this note, we shall give a different proof of their results, thus showing that their conclusions remain valid for infinite fields as well. We shall also show that the above-mentioned embedding of \({DQ}^-(2n + 1,\mathbb{K})\) into \({PG}(2^n -1,\mathbb{K}’)\) is polarized.